Solve 0=-1/20x^4+11/10x^2+45/20

$$0 = - \frac { 1 } { 20 } x ^ { 4 } + \frac { 11 } { 10 } x ^ { 2 } + \frac { 45 } { 20 }$$

Solve for x (complex solution)

$$x=i\sqrt{\sqrt{166}-11}\approx 1.372624758i$$

$$x=-i\sqrt{\sqrt{166}-11}\approx -0-1.372624758i$$

$$x=-\sqrt{\sqrt{166}+11}\approx -4.887136045$$

$$x=\sqrt{\sqrt{166}+11}\approx 4.887136045$$

Steps Using the Quadratic Formula

Reduce the fraction \frac{45}{20} to lowest terms by extracting and canceling out 5.

$$0=-\frac{1}{20}x^{4}+\frac{11}{10}x^{2}+\frac{9}{4}$$

Swap sides so that all variable terms are on the left hand side.

$$-\frac{1}{20}x^{4}+\frac{11}{10}x^{2}+\frac{9}{4}=0$$

Substitute t for x^{2}.

$$-\frac{1}{20}t^{2}+\frac{11}{10}t+\frac{9}{4}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute -\frac{1}{20} for a, \frac{11}{10} for b, and \frac{9}{4} for c in the quadratic formula.

$$t=\frac{-\frac{11}{10}±\sqrt{\left(\frac{11}{10}\right)^{2}-4\left(-\frac{1}{20}\right)\times \frac{9}{4}}}{-\frac{1}{20}\times 2}$$

Do the calculations.

$$t=\frac{-\frac{11}{10}±\frac{1}{10}\sqrt{166}}{-\frac{1}{10}}$$

Solve the equation t=\frac{-\frac{11}{10}±\frac{1}{10}\sqrt{166}}{-\frac{1}{10}} when ± is plus and when ± is minus.

$$t=11-\sqrt{166} t=\sqrt{166}+11$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for each t.

$$x=-i\sqrt{-\left(11-\sqrt{166}\right)} x=i\sqrt{-\left(11-\sqrt{166}\right)} x=-\sqrt{\sqrt{166}+11} x=\sqrt{\sqrt{166}+11}$$

Solve for x

$$x=-\sqrt{\sqrt{166}+11}\approx -4.887136045$$

$$x=\sqrt{\sqrt{166}+11}\approx 4.887136045$$

Steps Using the Quadratic Formula

Reduce the fraction \frac{45}{20} to lowest terms by extracting and canceling out 5.

$$0=-\frac{1}{20}x^{4}+\frac{11}{10}x^{2}+\frac{9}{4}$$

Swap sides so that all variable terms are on the left hand side.

$$-\frac{1}{20}x^{4}+\frac{11}{10}x^{2}+\frac{9}{4}=0$$

Substitute t for x^{2}.

$$-\frac{1}{20}t^{2}+\frac{11}{10}t+\frac{9}{4}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute -\frac{1}{20} for a, \frac{11}{10} for b, and \frac{9}{4} for c in the quadratic formula.

$$t=\frac{-\frac{11}{10}±\sqrt{\left(\frac{11}{10}\right)^{2}-4\left(-\frac{1}{20}\right)\times \frac{9}{4}}}{-\frac{1}{20}\times 2}$$

Do the calculations.

$$t=\frac{-\frac{11}{10}±\frac{1}{10}\sqrt{166}}{-\frac{1}{10}}$$

Solve the equation t=\frac{-\frac{11}{10}±\frac{1}{10}\sqrt{166}}{-\frac{1}{10}} when ± is plus and when ± is minus.

$$t=11-\sqrt{166} t=\sqrt{166}+11$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for positive t.

$$x=\sqrt{\sqrt{166}+11} x=-\sqrt{\sqrt{166}+11}$$